cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A226495 The number of primes of the form i^2+j^4 (A028916) <= 10^n, counted with multiplicity.

Original entry on oeis.org

2, 8, 34, 134, 615, 2813, 13415, 65162, 323858, 1626844, 8268241, 42417710
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2+j^4.
Primes with more than one representation are counted multiple times.
If we do not count repetitions, the sequence is A226497: 2, 6, 28, 121, 583, 2724, 13175, 64551, ..., .

Examples

			2 = 1^2+1^4, 5 = 2^2+1^4, 17 = 4^2+1^4 = 1^2+2^4, …, 97 = 9^2+2^4 = 4^2+3^4, etc.

Crossrefs

Cf. A028916, A226496, A226497, A226498.

Programs

  • Mathematica
    mx = 10^12; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[ lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[lst, # < 10^n &], {n, 12}]

A226496 The number of primes of the form i^2 + j^4 (A028916) <= 2^n, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 9, 13, 21, 34, 50, 77, 121, 191, 292, 458, 727, 1164, 1840, 2904, 4650, 7429, 11869, 19087, 30760, 49474, 79971, 129226, 209823, 340347, 552722, 898655, 1461698, 2381041, 3883079, 6338935, 10357549, 16935173, 27712338, 45381521, 51559329
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2 + j^4.
Counted with double representations. If we do not count doubles, the sequence is A226498.

Examples

			2 = 1^2+1^4, 5 = 2^2+1^4, 17 = 4^2+1^4 = 1^2+2^4, ..., 97 = 9^2+2^4 = 4^2+3^4, etc.

Crossrefs

Cf. A028916, A226495, A226497, A226498.

Programs

  • Mathematica
    mx = 2^40; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[ lst, a]], {i, Sqrt[mx]}, {j, Sqrt[ Sqrt[mx - i^2]]}]; Table[ Length@ Select[lst, # <2^n &], {n, 40}]

A226497 The number of primes of the form i^2+j^4 (A028916) <= 10^n.

Original entry on oeis.org

2, 6, 28, 121, 583, 2724, 13175, 64551, 322110, 1621929, 8254127
Offset: 1

Views

Author

Marek Wolf and Robert G. Wilson v, Jun 09 2013

Keywords

Comments

Even if a prime has more than one representation in the form i^2+j^4, it is only counted once.
Iwaniec and Friedlander have proved there is infinity of the primes of the form i^2+j^4.

Crossrefs

Cf. A028916, A226495, A226496 & A226498.

Programs

  • Mathematica
    mx = 2^40; lst = {};  Do[a = i^2 + j^4; If[ PrimeQ[a], AppendTo[lst, a]], {i, Sqrt[mx]}, {j, Sqrt[Sqrt[mx - i^2]]}]; Table[ Length@ Select[ Union@ lst, # < 10^n &], {n, 12}]
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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